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Transcript
1 Statistical Mechanics and MultiScale Simulation Methods ChBE 591-009 Prof. C. Heath Turner Lecture 15 • Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu • Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm 2 Review and Preview MD of hard disks • intuitive • collision detection and impulsive dynamics Monte Carlo • convenient sampling of ensembles • no dynamics • biasing possible to improve performance MD of soft matter • • • • • equations of motion integration schemes evaluation of dynamical properties extensions to other ensembles focus on atomic systems for now (deal with molecules later) 3 Classical Equations of Motion Several formulations are in use (see other slides on website) • Newtonian • Lagrangian • Hamiltonian Advantages of non-Newtonian formulations • • • • more general, no need for “fictitious” forces better suited for multiparticle systems better handling of constraints can be formulated from more basic postulates Assume conservative forces F U Gradient of a scalar potential energy 4 Newtonian Formulation Cartesian spatial coordinates ri = (xi,yi,zi) are primary variables • for N atoms, system of N 2nd-order differential equations d 2ri m 2 mri Fi dt Sample application: 2D motion in central force field yf y mx F eˆ x f (r )rˆ eˆ x xf x2 y 2 my F eˆ y f (r )rˆ eˆ y x2 2 • Polar coordinates are more natural and convenient mr 2 constant angular momentum mr f (r ) 2 mr 3 r fictitious (centrifugal) force F f (r )rˆ 5 Phase Space (again) Return to the complete picture of phase space • full specification of microstate of the system is given by the values of all positions and all momenta of all atoms G = (pN,rN) • view positions and momenta as completely independent coordinates connection between them comes only through equation of motion Motion through phase space • helpful to think of dynamics as “simple” movement through the highdimensional phase space facilitate connection to quantum mechanics basis for theoretical treatments of dynamics understanding of integrators G 6 Integration Algorithms Equations of motion in cartesian coordinates dr j dt dp j dt pj r (rx , ry ) m p ( px , p y ) Fj 2-dimensional space (for example) N F j Fij pairwise additive forces i 1 i j Desirable features of an integrator • • • • • • F minimal need to compute forces (a very expensive calculation) good stability for large time steps good accuracy conserves energy and momentum time-reversible More on these later area-preserving (symplectic) 7 Verlet Algorithm 1. Equations Very simple, very good, very popular algorithm Consider expansion of coordinate forward and backward in time 1 r (t ) t 3 O ( t 4 ) r (t t ) r (t ) m1 p(t ) t 21m F(t ) t 2 3! 1 r (t ) t 3 O ( t 4 ) r (t t ) r (t ) m1 p(t ) t 21m F(t ) t 2 3! Add these together r (t t ) r (t t ) 2r (t ) m1 F(t ) t 2 O( t 4 ) Rearrange r (t t ) 2r (t ) r (t t ) m1 F(t ) t 2 O( t 4 ) • update without ever consulting velocities! 8 Verlet Algorithm 2. Flow diagram One MD Cycle Configuration r(t) Previous configuration r(t-t) One force evaluation per time step Entire Simulation Initialization Compute forces F(t) on all atoms using r(t) Reset block sums New configuration 1 move per cycle Advance all positions according to r(t+t) = 2r(t)-r(t-t)+F(t)/m t2 Add to block sum cycles per block Compute block average blocks per simulation Add to block sum No End of block? Compute final results Yes Block averages 9 Verlet Algorithm 2. Flow Diagram t-t r v t t+t Compute new position from present and previous positions, and present force F Schematic from Allen & Tildesley, Computer Simulation of Liquids 10 Forces 1. Formalism Force is the gradient of the potential F21 u (r12 ) u (r12 ) u (r12 ) e ey Force on 1, x x1 y1 due to 2 du (r12 ) r12 r12 e ey x dr12 x1 y1 f (r ) 12 x12e x y12e y r12 F21 F12 x12 y12 2 1 r12 1/ 2 r12 ( x2 x1 ) 2 ( y2 y1 ) 2 4 Energy Force 2 0 -2 1.0 1.5 2.0 Separation, r/s 2.5 11 Forces 2. LJ Model Force is the gradient of the potential F21 f (r12 ) x12e x y12e y r12 x12 y12 2 e.g., Lennard-Jones model 1 r12 1/ 2 r12 ( x2 x1 ) 2 ( y2 y1 ) 2 s 12 s 6 u (r ) 4 r r 4 du f (r ) dr 13 7 2 48 s 1s s r 2 r 0 14 8 48 s 1 s F21 2 x12e x y12e y 2 r12 s r12 -2 Energy Force 1.0 1.5 2.0 Separation, r/s 2.5 12 Verlet Algorithm. 4. Loose Ends Initialization • how to get position at “previous time step” when starting out? • simple approximation r(t0 t ) r(t0 ) v(t0 ) t Obtaining the velocities • not evaluated during normal course of algorithm • needed to compute some properties, e.g. temperature diffusion constant • finite difference v(t ) 1 r(t t ) r(t t ) O( t 2 ) 2 t 13 Verlet Algorithm 5. Performance Issues Time reversible • forward time step r (t t ) 2r (t ) r (t t ) m1 F(t ) t 2 • replace t with t r (t ( t )) 2r (t ) r (t ( t )) m1 F(t )( t ) 2 r (t t ) 2r (t ) r (t t ) m1 F (t ) t 2 • same algorithm, with same positions and forces, moves system backward in time Numerical imprecision of adding large/small numbers O(t1) O(t1) r (t t ) r (t ) r (t ) r (t t ) m1 F(t ) t 2 O(t0) O(t0) O(t2) 14 Initial Velocities (from Lecture 3) Random direction • randomize each component independently • randomize direction by choosing point on spherical surface Magnitude consistent with desired temperature. Choices: • • • • Maxwell-Boltzmann: prob(vx ) exp 12 mvx2 / kT 2 Uniform over (-1/2,+1/2), then scale so that N1 vi, x kT / m Constant at vx kT / m Same for y, z components Be sure to shift so center-of-mass momentum is zero Px N1 pi, x pi, x pi, x Px 15 Leapfrog Algorithm Eliminates addition of small numbers O(t2) to differences in large ones O(t0) Algorithm r (t t ) r (t ) v(t 12 t ) t v(t 12 t ) v(t 12 t ) m1 F(t ) t Mathematically equivalent to Verlet algorithm r (t t ) r (t ) v(t 12 t ) m1 F(t ) t t r(t) as evaluated from r(t ) r(t t ) v(t 12 t ) t previous time step 16 Leapfrog Algorithm 2. Flow Diagram t-t r v t t+t Compute velocity at next half-step F Schematic from Allen & Tildesley, Computer Simulation of Liquids 17 Leapfrog Algorithm 2. Flow Diagram t-t r t t+t Compute next position v F Schematic from Allen & Tildesley, Computer Simulation of Liquids 18 Leapfrog Algorithm. 3. Loose Ends Initialization • how to get velocity at “previous time step” when starting out? • simple approximation v(t0 t ) v(t0 ) m1 F(t0 ) 12 t Obtaining the velocities • interpolate 1 v (t ) v (t 12 t ) v (t 12 t ) 2 19 Velocity Verlet Algorithm Roundoff advantage of leapfrog, but better treatment of velocities Algorithm r (t t ) r (t ) v(t ) t 21m F(t ) t 2 v(t t ) v(t ) 21m F(t ) F(t t ) t Implemented in stages • • • • • evaluate current force compute r at new time add current-force term to velocity (gives v at half-time step) compute new force add new-force term to velocity Also mathematically equivalent to Verlet algorithm (in giving values of r) 20 Velocity Verlet Algorithm 2. Flow Diagram t-t r t t+t Compute new position v F Schematic from Allen & Tildesley, Computer Simulation of Liquids 21 Velocity Verlet Algorithm 2. Flow Diagram t-t r t t+t Compute velocity at half step v F Schematic from Allen & Tildesley, Computer Simulation of Liquids 22 Velocity Verlet Algorithm 2. Flow Diagram t-t r t t+t Compute force at new position v F Schematic from Allen & Tildesley, Computer Simulation of Liquids 23 Velocity Verlet Algorithm 2. Flow Diagram t-t r t t+t Compute velocity at full step v F Schematic from Allen & Tildesley, Computer Simulation of Liquids 24 Other Algorithms Predictor-Corrector • not time reversible • easier to apply in some instances constraints rigid rotations Beeman • better treatment of velocities Velocity-corrected Verlet 25 Summary Several formulations of mechanics • Hamiltonian preferred independence of choice of coordinates emphasis on phase space Integration algorithms • Calculation of forces • Simple Verlet algorithsm Verlet Leapfrog Velocity Verlet Next up: Calculation of dynamical properties